σ —¦ T · . One direction requires showing that for ±: T ’ T and any object A,

’

±A = µ A —¦ ±T A —¦ T · A. This follows from the commutativity of the following

diagram, in which the square commutes because ± is a natural transformation,

the triangle commutes by de¬nition and the bottom row is the identity by the

de¬nition of triple.

T · AE

TA TT A

d

d

(9)

·T A

±A d

d

d

‚

c c

ETTA ETA

TA

T ·A µA

The other direction is more complicated. Suppose we are given V . We must show

that for any T -algebra (A, a),

V (A, a) = (A, a —¦ σA —¦ T · A) (10)

where by de¬nition σA is the T-algebra structure on T A obtained by applying

V to the free algebra (T A, µ A).

In the ¬rst place,

µ A: (T 2 A, µ T A) ’ (T , µ A)

’

is a morphism of T -algebras, so because of (7), µ is also a morphism of the

T-algebras (T 2 A, σT A) ’ (T A, σA). This says that the square in the diagram

’

below commutes. Since the triangle commutes by de¬nition of triple, (10) is true

at least for images under T of free T -algebras.

T · TE

A σT A

ET T A

TT A TT T A

d

d

(11)

Tµ A µA

idd

d

dc

d

‚ c

T T A σA E T A

Now by Proposition 4, Section 3.3, for any T -algebra (A, a),

·A

’ ’ ’ (T A, µ A) ’a (A, a)

’’

2

(T A, µ T A) ’ ’ ’ ’’ (12)

’’

TA

3.7 Adjoint Triples 135

is a U -contractible coequalizer diagram. Applying U ± (where ± = σ —¦ T · ) must

give a U -contractible coequalizer diagram since U ± commutes with the underlying

functors. Because (10) is true of images of free algebras, that diagram is

·A

’ ’ ’ (T A, σA) ’a (A, a)

’’

2

(T A, σT A) ’ ’ ’ ’’ (13)

’’

TA

where b = a —¦ σA —¦ T · A. Since V also commutes with underlying functors,

applying V to (12) also gives a U -contractible coequalizer pair, with the same

left and middle joints as (13) (that is how σ was de¬ned). Its coequalizer must

be V (A, a) since the underlying functors create coequalizers. Thus (10) follows

as required.

Exercises 3.6

1. Show that for a given category C , the triples in C and their morphisms form

a category.

2. Show that U ± as de¬ned in the proof of Theorem 3 is a functor.

3. Let T be the Abelian group triple and T the free group triple. What is the

triple morphism ± corresponding to the inclusion of Abelian groups into groups

given by Theorem 3?

3.7 Adjoint Triples

In this section, we state and prove several theorems asserting the existence of

adjoints to certain functors based in one way or another on categories of triple

algebras. These are then applied to the study of the tripleability of functors

which have both left and right adjoints.

Induced Adjoints

Theorem 1. In the following diagram (not supposed commutative) of categories

and functors,

W EB

B

d

d

s

dd

F d dU F U

dd

d d

‚©

C

136 3 Triples

suppose that

(i) F is left adjoint to U ,

(ii) F is left adjoint to U ,

(iii) W F is naturally isomorphic to F ,

(iv) U is tripleable, and

(v) W preserves coequalizers of U -contractible pairs.

Then W has a right adjoint R for which U R ∼ U .

=

Proof. We will de¬ne R by using Corollary 2 of Section 3.6. Let the triples cor-

responding to the adjunctions be T = (T, ·, µ) and T = (T , · , µ ) respectively.

As usual, suppose that B = C T and U = U T . De¬ne σ: T T ’ T so that

’

∼

=E

U WUF U F UF

σ

UW F

c c

EUF

U WF ∼

=

commutes.

Applying U W to diagram (4) of Section 3.1 and evaluating at F gives a

diagram which, when the isomorphism of (iii) is applied, shows that σ —¦ σT =

σ —¦T µ. An analogous (easier) proof using (2) of Section 3.1 shows that σ —¦T · = id.

Thus by Corollary 2 of Section 3.6, ± = σ —¦ · T : T ’ T is a morphism of triples.

’

The required functor R is B ’ C T ’ C T , where the ¬rst arrow is the

’ ’

comparison functor and the second is the functor V induced by ±.

For an object B of B , we have, applying the de¬nitions of R, the comparison

functor ¦ for U , and the functor V determined by ±, the following calculation:

U RB = U V ¦B = U V (U B , U B )

= U (U B , U B —¦ ±U B ) = U B ,

so U R = U as required.

We now show that R is right adjoint to W insofar as free objects are concerned,

and then appeal to the fact that algebras are coequalizers. (Compare the proof

of Theorem 9 of Section 3.3). The following calculation does the ¬rst: For C an

object of C and B an object of B ,

Hom(W F C, B ) ∼ Hom(F C, B )

=

∼ Hom(C, U B ) ∼ Hom(C, U RB ) ∼ Hom(F C, RB )—¦

= = =

3.7 Adjoint Triples 137

Now any object of B has a presentation

F C2 ’ F C1 ’ B

’ ’ (—)

’’

by a U -contractible coequalizer diagram. Since U is tripleable, (—) is a coequalizer.

Thus the bottom row of the diagram

E

E Hom(W F C1 , B )

Hom(W B, B ) E Hom(W F C2 , B )

c c

E

E Hom(F C1 , RB )

Hom(B, RB ) E Hom(F C2 , RB )

is an equalizer. By (v), W preserves coequalizers of U - contractible pairs, so the

top row is an equalizer. The required isomorphism of Homsets follows.

Theorem 2. In the following diagram (not supposed commutative) of categories

and functors,

W EB

B

d

s d

dd

F d dU F U

dd

d

‚©

d

C

(a) Suppose that

(i) F is left adjoint to U ,

(ii) F is left adjoint to U ,

(iii) W F is naturally isomorphic to F ,

(iv) U is of descent type, and

(v) B has and W preserves coequalizers of U -contractible coequalizer pairs.

Then W has a right adjoint R for which U R ∼ U .

=

(b) Suppose that

(i) F is left adjoint to U ,

(ii) F is left adjoint to U ,

(iii) U W is naturally isomorphic to U ,

(iv) U is of descent type, and

(v) B has coequalizers.

Then W has a left adjoint L for which LF ∼ F .

=

Proof.

138 3 Triples

(a) We must show that the hypothesis in Theorem 1 that U is tripleable can

be weakened to the assumption that it is of descent type. Consider the diagram

Ψ E W EB

CT EB

¦

s

dd T

dd

dd

F TddU T F U F U

dd

dd

d c

‚ ©

d

C

in which the left adjoint Ψ exists because B has coequalizers (Proposition 11,

Section 3.3).

Theorem 1 implies that W —¦ Ψ has a right adjoint S: B ’ C T . To apply

’

T ∼ F . This follows from the given

Theorem 1 we need to know that W —¦ Ψ —¦ F =

fact W —¦ F ∼ F and the fact that the counit of the adjunction between ¦ and Ψ

=

must be an isomorphism by Exercise 12 of Section 1.9.

From this we have for objects A of C T and B of B that

Hom(¦ΨA, SB ) ∼ Hom(W Ψ¦ΨA, B )

=

∼ Hom(W ΨA, B ) ∼ Hom(A, SB )—¦

= =

Thus by Exercise 7(c) of Section 1.9, every object of the form SB is ¦B for some

B in B. Since ¦ is full and faithful, this allows the de¬nition of a functor R: B

’ B for which S = ¦ —¦ W .

’

The following calculation then shows that R is right adjoint to W :

Hom(W B, B ) ∼ Hom(W Ψ¦B, B ) ∼ Hom(¦B, SB )

= =

∼ Hom(¦B, ¦RB ) ∼ Hom(B, RB )—¦

= =

(b) If F C is an object in the image of F , then we have

Hom(F C, W B) ∼ Hom(C, U W B)

=

∼ Hom(C, U B) ∼ Hom(F C, B)

= =

which shows that FC represents the functor Hom(F C, W ’). Moreover, the

Yoneda lemma can easily be used to show that maps in B between objects in the

image of F give rise to morphisms in B with the required naturality properties.

Thus we get a functor L de¬ned at least on the full subcategory whose objects

are the image of F . It is easily extended all of B by letting

F C2 ’ F C1 ’ B

’ ’

’’

3.7 Adjoint Triples 139

be a coequalizer and de¬ning LB so that